On the Linear Complexity of Binary Sequences of Period $4N$ With Optimal Autocorrelation Value/Magnitude

Three classes of binary sequences of period 4
N
with optimal autocorrelation value/magnitude have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period
N
, i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the properties of roots of the corresponding sequence polynomials of the interleaved sequences with period 4
N
and optimal autocorrelation value/magnitude are discussed in the splitting field of
xN
-1 . As a consequence, both the minimal polynomials and linear complexities of these three classes of sequences are completely determined except for the case of the sequences obtained from the generalized GMW sequences. For the latter, the minimal polynomial and linear complexity can be specially obtained if the sequence is constructed based on
m
-sequences instead of generalized GMW sequences.